Browsing by Author "Moll, Victor H."

A new generalization of the modified Bessel function of the second kind Kz(x) is studied. Elegant series and integral representations, a differential-difference equation and asymptotic expansions are obtained for it thereby anticipating a rich theory that it may possess. The motivation behind introducing this generalization is to have a function which gives a new pair of functions reciprocal in the Koshliakov kernel cos⁡(πz)M2z(4x)−sin⁡(πz)J2z(4x) and which subsumes the self-reciprocal pair involving Kz(x). Its application towards finding modular-type transformations of the form F(z,w,α)=F(z,iw,β), where αβ=1, is given. As an example, we obtain a beautiful generalization of a famous formula of Ramanujan and Guinand equivalent to the functional equation of a non-holomorphic Eisenstein series on SL2(Z). This generalization can be considered as a higher level analogue of the general theta transformation formula. We then use it to evaluate an integral involving the Riemann Ξ-function and consisting of a sum of products of two confluent hypergeometric functions.

In 1998 Don Zagier introduced the modified Bernoulli numbers Bn∗ and showed that they satisfy amusing variants of some properties of Bernoulli numbers. In particular, he studied the asymptotic behavior of B2n∗, and also obtained an exact formula for them, the motivation for which came from the representation of B2 n in terms of the Riemann zeta function ζ(2 n). The modified Bernoulli numbers were recently generalized to Zagier polynomials Bn∗(x). For 0 < x< 1 , an exact formula for B2n∗(x) involving infinite series of Bessel function of the second kind and Chebyshev polynomials, that yields Zagier’s formula in a limiting case, is established here. Such series arise in diffraction theory. An analogous formula for B2n+1∗(x) is also presented. The 6-periodicity of B2n+1∗ is deduced as a limiting case of it. These formulas are reminiscent of the Fourier expansions of Bernoulli polynomials. Some new results, for example, the one yielding the derivative of the Bessel function of the first kind with respect to its order as the Fourier coefficient of a function involving Chebyshev polynomials, are obtained in the course of proving these exact formulas. The asymptotic behavior of Zagier polynomials is also derived from them. Finally, a Zagier-type exact formula is obtained for B2n∗(-32)+B2n∗.